Logic System Proofs

The claim "System X is System Y plus the axiom Z" invites
the question of "Why?". There are two basic ways of answering
this sort of question. The first, which I usually do, is to
cite some authority on the subject. The second is to actually
prove it, which I don't (yet) generally do..

This section is where I intend to accumulate the proofs.

[As long as this is getting automated, automated proofs
would be a great help here.]